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^
z
l 3
^
z
x 3
^
x
π/2+I
^
^
x
x
N
x 1
S
N
I
I
B 0
B 0
^
z
Fig. 7.1. Two chosen Cartesian systems for the Northern Hemisphere. The system
( x, y, z ) is Cartesian-altitude coordinate system and ( x ,y z ) is connected with the
geomagnetic field. I is the inclination angle
Figure 7.1). Dip angle I< 0 in the Southern Hemisphere and I> 0inthe
Northern Hemisphere. Input two Cartesian coordinate systems: the first one
{
x ,y ,z }
related to the main magnetic field B 0 and another one
{
x, y, z
}
is
x , y , z
a Cartesian-altitude system. Let
be unit
vectors of the x -, y -, z -and x -, y -, z - axes, respectively. Let the z = 0 plane
be the boundary between the conducting ionosphere and the atmosphere. Al-
though this boundary is not exactly determined, this inaccuracy does not lead
to any practical diculties. We neglect the curvature of the ground surface
and match it with the plane z =
{
= B 0 /B 0 }
and
{
x , y , z
}
x ,y ,z }
h . Coordinate system
{
turns out
from
by the rotation with respect to axis y at an angle π/ 2+ I .The
x and y axes are parallel to the ionosphere and ground surface. The x axis
is southwards, y is eastwards and z axis is vertically upwards. Let us direct
axis z along B 0 and axis y eastwards. Set also an oblique coordinate sys-
{
x, y, z
}
tem x 1 ,x 2 ,x 3 with horizontal surfaces x 3 = const and coordinate lines x 3
coincident with the field-lines.
The orthogonal coordinate system related to the magnetic field for the
Northern Hemisphere is shown in the left frame of Fig. 7.1 and the oblique
coordinate system is shown in the right frame. The coordinate systems are
related as
x sin I
z cos I,
x =
x =
x sin I + z cos I,
y = y ,
y = y,
z = x cos I
z sin I,
z =
x cos I
z sin I,
(7.1)
x = x 1 + x 3 cot I,
x 1 = x
z cot I,
y = x 2 ,
x 2 = y,
z = x 3 ,
x 3 = z,
(7.2)
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